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Quantum Resistant Authentication System for Authorizing Private Key Signing

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Introduction

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Public Keys

coin1hj5fveer5cjtn4wd6wstzugjfdxzl0xpdpsrzs

• Think of the public key as your digital signature. It lets others recognize you without sharing your personal details, fostering trust and openness in every exchange.

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Private Keys

123SomePr1vat3Key...

• The private key is a secret component used to generate digital signatures, ensuring that only the authorized owner can initiate transactions or communications, safeguarding against unauthorized access.

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Quantum One–Time Pad (QOTP) and ENIGMA Cipher Integration

• A–Z (26 characters)
• 0–9 (10 characters)
• 艾勒, 哦, 维, 衣 (4 symbols)

• Red → UP
• Green → DOWN
• Blue → LEFT
• Yellow → RIGHT

• The QOTP’s entropy-driven rotation assigns colors to characters based on their position in the pad, which is synchronized with ENIGMA’s hyperplane mapping. This dual-system approach ensures quantum resistance by leveraging dynamic, unpredictable rotations and a private, static mapping, making it computationally infeasible for adversaries to predict or reverse-engineer the sequence.

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Integrated QOTP and ENIGMA Operation: Example with “LOVE”

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Step 1: Universal Language ($\Lambda$)

• The universal language $\Lambda$ comprises 40 distinct symbols, forming a comprehensive alphabet for both QOTP and ENIGMA. This diverse set ensures a large keyspace, enhancing cryptographic strength by maximizing the possible combinations for subkeys and mappings.

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Step 2: Private Key ($\Omega$) and Subkeys ($\Omega_i$)

• The private key $\Omega$ is an ordered sequence of subkeys $\Omega_i$, each drawn from $\Lambda$. In the QOTP system, each subkey’s position in the entropy-rotated pad determines its color and initial grouping, while in the ENIGMA system, these subkeys are located within specific hyperplanes ($\Gamma_i$) during each round of the $\Sigma$-Interactive Protocol. The term “position” refers to the character’s location in the QOTP pad sequence, which influences its hyperplane assignment in ENIGMA. For example, $\mathrm{L}$ at position 11 falls into the yellow hyperplane ($\Gamma_1$), mapped to RIGHT via $\Psi$. This dual-system approach ensures that each subkey’s cryptographic role is dynamically determined by entropy-based rotations and securely mapped within the holographic manifold, enhancing both visual clarity and quantum-resistant security.

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Step 3: Secret Synonym Mapping ($\Psi$)

• $\Gamma_1$ (Yellow) → $\Xi_1$ (RIGHT)
• $\Gamma_2$ (Green) → $\Xi_2$ (DOWN)
• $\Gamma_3$ (Blue) → $\Xi_3$ (LEFT)
• $\Gamma_4$ (Red) → $\Xi_4$ (UP)

• The mapping $\Psi$ is a static, private function that associates each hyperplane $\Gamma_i$ with a bearing direction $\Xi_i$, known only to the observer. This mapping aligns with the QOTP’s color-coded system, ensuring that each subkey’s direction is both visually intuitive and cryptographically secure, maintaining zero-knowledge properties.

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Step 4: Commitment ($\Gamma$)

• The commitment $\Gamma$ binds the private key $\Omega$ and the hashed mapping $H(\Psi)$ using a quantum-resistant hash function $H$ (e.g., SHAKE256). The $\parallel$ operator denotes concatenation, ensuring that any alteration to $\Omega$ or $\Psi$ invalidates $\Gamma$, providing integrity and authenticity.

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Step 5: Möbius Rotation and $\Sigma$-Interactive Protocol with “LOVE”

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Round 1 (Initial Pad, $\Omega_1 = \mathrm{L}$)

• Red (UP): A, E, I, M, Q, U, Y, 2, 6, 艾勒
• Green (DOWN): B, F, J, N, R, V, Z, 3, 7, 哦
• Blue (LEFT): C, G, K, O, S, W, 0, 4, 8, 维
• Yellow (RIGHT): D, H, L, P, T, X, 1, 5, 9, 衣 Locate $\Omega_1 = \mathrm{L}$: Found in $\Gamma_1$ (Yellow).
  Witness: $\Psi(\Gamma_1) = \Xi_1$ → RIGHT.
  Commitment:

• The initial pad groups characters by color, with $\mathrm{L}$ located in the yellow hyperplane ($\Gamma_1$). The mapping $\Psi$ assigns the direction RIGHT, and the commitment hashes $\mathrm{L}$, RIGHT, and a nonce $r_1$, ensuring secure, verifiable authentication.

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Round 2 (Rotation by Offset 4, $\Omega_2 = \mathrm{O}$)

• Red (UP): E, I, M, Q, U, Y, 2, 6, 艾勒, A
• Green (DOWN): F, J, N, R, V, Z, 3, 7, 哦, B
• Blue (LEFT): G, K, O, S, W, 0, 4, 8, 维, C
• Yellow (RIGHT): H, L, P, T, X, 1, 5, 9, 衣, D Locate $\Omega_2 = \mathrm{O}$: Found in $\Gamma_3$ (Blue).
  Witness: $\Psi(\Gamma_3) = \Xi_3$ → LEFT.
  Commitment:

• After a 4-position offset, $\mathrm{O}$ shifts to the blue hyperplane ($\Gamma_3$), mapped to LEFT. The commitment hashes $\mathrm{O}$, LEFT, and $r_2$, maintaining zero-knowledge security through dynamic rotation.

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Round 3 (Rotation by Offset 8, $\Omega_3 = \mathrm{V}$)

• Red (UP): I, M, Q, U, Y, 2, 6, 艾勒, A, E
• Green (DOWN): J, N, R, V, Z, 3, 7, 哦, B, F
• Blue (LEFT): K, O, S, W, 0, 4, 8, 维, C, G
• Yellow (RIGHT): L, P, T, X, 1, 5, 9, 衣, D, H Locate $\Omega_3 = \mathrm{V}$: Found in $\Gamma_2$ (Green).
  Witness: $\Psi(\Gamma_2) = \Xi_2$ → DOWN.
  Commitment:

• An 8-position offset places $\mathrm{V}$ in the green hyperplane ($\Gamma_2$), mapped to DOWN. The commitment hashes $\mathrm{V}$, DOWN, and $r_3$, ensuring the integrity of the authentication process.

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Round 4 (Rotation by Offset 12, $\Omega_4 = \mathrm{E}$)

• Red (UP): M, Q, U, Y, 2, 6, 艾勒, A, E, I
• Green (DOWN): N, R, V, Z, 3, 7, 哦, B, F, J
• Blue (LEFT): O, S, W, 0, 4, 8, 维, C, G, K
• Yellow (RIGHT): P, T, X, 1, 5, 9, 衣, D, H, L Locate $\Omega_4 = \mathrm{E}$: Found in $\Gamma_4$ (Red).
  Witness: $\Psi(\Gamma_4) = \Xi_4$ → UP.
  Commitment:

• A 12-position offset shifts $\mathrm{E}$ to the red hyperplane ($\Gamma_4$), mapped to UP. The commitment hashes $\mathrm{E}$, UP, and $r_4$, completing the secure mapping for “LOVE”.

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Step 6: Final Witness Sequence for “LOVE”

• The final witness sequence $\Xi$ encapsulates the bearing directions for each subkey, derived from the hyperplane mappings and QOTP rotations. This sequence is visually represented with color-coded characters and direction icons, ensuring clarity and alignment with the cryptographic process.

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Step 7: Verification

• The verification function $\Theta$ evaluates the commitments $\text{Comm}_i$, challenges $e_i$, and responses $s_i$ for each round, leveraging zero-knowledge proof principles. This ensures that the observer’s knowledge of $\Omega_i$ and $\Psi$ remains confidential, with no leakage of sensitive information, even under quantum attacks.

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Summary of Equations Used

1. Universal Language:

• The universal language $\Lambda$ defines a 40-character alphabet, encompassing letters, digits, and symbols, forming the foundation for key generation and mapping. This expansive set maximizes entropy, ensuring robust cryptographic strength.

2. Private Key:

• The private key $\Omega$ is an ordered sequence of subkeys $\Omega_i$, each drawn from $\Lambda$. This structure allows for flexible, dynamic key generation, with each subkey’s position in the QOTP pad determining its cryptographic role.

3. Secret Synonym Mapping (Holographic Morphism):

• The mapping $\Psi$ is a private, static function that associates each hyperplane $\Gamma_i$ with a bearing direction $\Xi_i$. This mapping, known only to the observer, aligns with QOTP’s color-coded system, ensuring that each subkey’s direction is both visually intuitive and cryptographically secure.

4. Commitment:

• The commitment $\Gamma$ binds the private key $\Omega$ and the hashed mapping $H(\Psi)$ using a quantum-resistant hash function $H$ (e.g., SHAKE256). The \parallel operator denotes concatenation, ensuring that any alteration to $\Omega$ or $\Psi$ invalidates $\Gamma$, providing integrity and authenticity.

5. $\Sigma$-Interactive Protocol Steps (Per Round):

• • Commitment: Hashes the subkey $\Omega_i$, witness $\Xi_i$, and nonce $r_i$ using $H$, creating a secure, verifiable commitment.
  • Response: Computes $s_i$ based on $\Omega_i$, $\Xi_i$, challenge $e_i$, and $r_i$, ensuring the observer’s knowledge is verifiable without revealing secrets.
  • Verification: Uses $\Theta$ to validate the commitment, response, and challenge, enforcing zero-knowledge properties and ensuring no information leakage.

6. Witness Sequence:

• The witness sequence $\Xi$ encapsulates the bearing directions for each subkey, derived from hyperplane mappings and QOTP rotations. This sequence is the final output of the authentication process, ensuring alignment with the cryptographic framework.

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Symbol Definitions